Study of cortical bone fracture behaviour under fatigue loads

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Study of cortical bone fracture

behaviour under fatigue loads

Filipe Gonçalo Andrade da Silva

2020

Thesis presented to the Faculty of Engineering of the University

of Porto for the Doctor Degree in Mechanical Engineering

Supervisor: Marcelo Francisco de Sousa Ferreira de Moura

Co-Supervisor: José Joaquim Lopes Morais

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I wish to dedicate this thesis to all the scientific community the ancient the present and the imminent.

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Bone fractures are quite common. They can be derived from accidents, fatigue loading during walking or running, diseases or as result of administration of drugs for a long time. The study of bone fracture under fatigue loads is a very important research topic with a huge impact in human health. The purpose of this work is to develop methodologies that allow the determination of the fatigue laws under pure mode (I, II) and under mixed-mode I+II loading for bovine and human cortical bone tissue. In this context, fatigue testing using miniaturized versions of Double Cantilever Beam tests for mode I, End-Notched Flexure for mode II, and Single-Leg Bending for mixed-mode I+II are applied. Additionally, numerical analysis involving two cohesive zone models that simulate quasi-static fracture and fatigue/fracture behaviour under the analysed loading modes (I, II, I+II) were used, aiming to validate all the employed procedures.

The results of this work aim the development of tools to perform systematic studies on bone fracture behaviour under fatigue loads. Effectively, the proposed tests will allow the evaluation of the influence of parameters such as age, drugs consumption, pathologies, among others, on the performance of cortical bone tissue submitted to fatigue loads. These procedures can be viewed as being very useful on the definition of appropriate therapies for the resolution of health problems affecting fatigue behaviour of bone tissue.

Keywords: Cortical bone; Fatigue/fracture characterization; Mode I and mode II; Mixed-mode I+II; Cohesize zone analysis.

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As fraturas ósseas são comuns em consequência de quedas, doenças, envelhecimento e fenómenos de fadiga resultantes das atividades diárias (por exemplo, caminhar ou correr). Consequentemente, o estudo da fratura do osso sob solicitações de fadiga é um tópico de investigação extremamente importante com impacto muito relevante na saúde humana. Neste trabalho pretende-se desenvolver metodologias que permitam determinar as leis de fadiga sob solicitações de modo puro (I, II) e de modo misto (I+II) do tecido ósseo cortical de bovino e humano. Realizaram-se ensaios de fadiga usando versões miniaturizadas dos testes Double-Cantilever Beam (DCB) para o modo I, End-Notched

Flexure (ENF) para o modo II e Single-Leg Bending (SLB) para o modo-misto I+II.

Adicionalmente, efetuaram-se análises numéricas envolvendo dois modelos de dano coesivos que permitem simular o comportamento quase-estático à fratura e à fratura sob fadiga, considerando os diferentes modos de carregamento (I, II, I+II) com o objetivo de validar os procedimentos utilizados.

Os resultados deste trabalho permitirão o desenvolvimento de ferramentas para a execução de estudos sistematizados sobre o comportamento do tecido ósseo cortical sob solicitações de fadiga. Efetivamente, os testes propostos possibilitarão avaliar o efeito de parâmetros, tais como a idade, consumo de fármacos, existência de patologias, entre outros, no desempenho do tecido ósseo cortical sob solicitações de fadiga, constituindo-se assim como instrumentos muito úteis na definição de terapias adequadas à resolução de problemas de saúde que afetem o comportamento à fadiga do tecido ósseo cortical.

Palavras chave: Osso cortical; Caracterização à fratura sob fadiga; Modo I e modo II; Modo misto I+II; Análise com modelo coesivos.

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My first acknowledge is for “Fundação para a Ciência e a Tecnologia (FCT– Portugal)” for the conceded financial support through Grant number SFRH/BD/111516/2015, that was fundamental to perform this work. I will also thank to all institutions and centres of research that gave me all conditions and support to do the work: HVUTAD (Hospital Veterinário da UTAD), CITAB (Centre for the Research and Technology of Agro-Environmental and Biological Sciences), UTAD (Universidade de Trás-os-Montes e Alto Douro), INEGI (Instituto de Ciência e Inovação em Engenharia Mecânica e Engenharia Industrial) and FEUP (Faculdade de Engenharia da Universidade do Porto). Specifically, I thank for the help and support of Dra. Isabel Dias and HVUTAD (Hospital Veterinário da UTAD) that make possible to obtain the material (bovine cortical bone) and provide all the necessary supplies to prepare and properly package the samples.

I am grateful to my supervisor Marcelo Francisco de Sousa Ferreira de Moura and both co-supervisors José Joaquim Lopes Morais and Nuno Miguel Magalhães Dourado, for the orientations and opportunity to integrate this research team. This circumstance allowed acquiring diverse knowledge and many different experiences. Their orientations, advices and fruitful discussions were essential to obtain all the “know-how” necessary to achieve all goals of this work.

Great and special thanks for a list of researchers, whose help was crucial to design, build and programme the test machine necessary to the execution of the experimental work/tests under diverse conditions: Prof. José Reina, Prof. José Ramiro, Prof. Marcos Martins and all colleagues and investigators Fernando Silva, Jhonny Rodrigues, Tiago Rodrigues, Raul Moreira, Diogo Vale, Armanda Marques, Maria do Carmo and the master student Jorge Oliveira.

I would also like to thank the remaining members of our research team for all their help and commitment, whenever they were requested, Cristovão Santos, Fábio Pereira, Francis Ramírez and the master student Carlos Martins.

Finally, I would also thank to all my friends and Family for the encouraging words and support.

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During this thesis several publications were achieved concerning the

works produced in the curricular classes and works related with fracture

characterisation of bone.

1. F. Silva, M.F.S.F.de Moura, N. Dourado, J. Xavier, F.A.M. Pereira, J.J.L. Morais, M.I.R. Dias, “Mixed-mode I+II fracture characterization of human cortical bone using the Single Leg Bending test”, Journal of the Mechanical Behavior of Biomedical Materials, 54: 72–81, 2016

https://doi.org/10.1016/j.jmbbm.2015.09.004

Journal impact factor: 3.110; ENGINEERING, Biomedical Rank: 22/77 – Quartil 2

2. F.A.M.Pereira, M.F.S.F.de Moura, N.Dourado, J.J.L.Morais, F.G.A.Silva,

M.I.R.Dias, Bone fracture characterization under mixed-mode I + II loading using the MMB test, Engineering Fracture Mechanics, Vol. 166, October 2016, Pages 151-163

https://doi.org/10.1016/j.engfracmech.2016.08.011

Journal impact factor: 2.151; MECHANICS Rank: 41/133 Quartil 2

3. F.G.A. Silva, M.F.S.F. de Moura, A.G. Magalhães, Low velocity impact behaviour of a hybrid carbon-epoxy/cork laminate, (2017) Strain Vol. 53, Issue 6

https://doi.org/10.1111/str.12241

Journal impact factor: 1.320; ENGINEERING Rank: 79/128 Quartil 3

4. F.G.A. Silva, M.F.S.F. de Moura, N. Dourado, J. Xavier, F.A.M. Pereira, J.J.L. Morais, M.I.R. Dias, P. J. Lourenço, F.M. Judas, “Fracture characterization of human cortical bone under mode II loading using the end-notched flexure test”, Medical & Biological Engineering & Computing, (2017) 55: 1249

https://doi.org/10.1007/s11517-016-1586-6

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three-dimensional progressive mixed-mode I+II damage model”, Engineering Fracture Mechanics,Volume 218, 2019, 106585

https://doi.org/10.1016/j.engfracmech.2019.106585

Journal impact factor: 2.908; MECHANICS Rank: 27/134 Quartil 1 (2018)

6. F.G.A.Silva M.F.S.F.de Moura, R.D.F.Moreira,“Influence of adverse temperature and moisture conditions on the fracture behaviour of single-strap repairs of carbon-epoxy laminates” International Journal of Adhesion and Adhesives Volume 96, 2020, 102452.

https://doi.org/10.1016/j.ijadhadh.2019.102452

Journal impact factor: 2.501; ENGINEERING Rank: 50/138 Quartil 2 (2018)

Publications related with partnerships

7. M.F.S.F.de Moura, J.P.M.Gonçalves,F.G.A.Silva, A new energy based mixed-mode cohesive zone model, International Journal of Solids and Structures, Volumes 102– 103, 15 December 2016, Pages 112-119

https://doi.org/10.1016/j.ijsolstr.2016.10.012

8. M.F.S.F.de Moura, P.M.L.C.Cavaleiro, F.G.A.Silva, N.Dourado, Mixed-mode I+II fracture characterization of a hybrid carbon-epoxy/cork laminate using the Single-Leg Bending test, Composites Science and Technolog , Volume 141, 22 March 2017, Pages 24-31

https://doi.org/10.1016/j.compscitech.2017.01.001

Journal impact factor: 5.160; MATERIALS SCIENCE Rank: 1/26 Quartil 1 9. R.A.M.Santos, P.N.B.Reis, F.G.A.Silva, M.F.S.F.de Moura, Influence of inclined

holes on the impact strength of CFRP composites, Composite Structures, Vol. 172, 15 July 2017, Pages 130-136

https://doi.org/10.1016/j.compstruct.2017.03.086

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femtosecond laser treated surfaces on the mode I fracture toughness of carbon-epoxy bonded joints, International Journal of Adhesion and Adhesives, Vol. 82, April 2018, Pages 108-113

https://doi.org/10.1016/j.ijadhadh.2018.01.005

11. J.P.Reis, M.F.S.F.de Moura, F.G.A.Silva, N.Dourado, Dimensional optimization of carbon-epoxy bars for reinforcement of wood beams, Composites Part B:

Engineering, Vol. 139, 2018, Pages 163-170 https://doi.org/10.1016/j.compositesb.2017.11.046

Journal impact factor: 6.864; MATERIALS SCIENCE Rank: 1/88 Quartil 1

12. N.Dourado, F.G.A.Silva, M.F.S.F.de Moura, Fracture behavior of wood-steel dowel joints under quasi-static loading, Construction and Building Materials, Vol. 176, 2018, Pages 14-23

https://doi.org/10.1016/j.conbuildmat.2018.04.230

13. R.D.F.Moreira, V.Oliveira, F.G.A.Silva, R.Vilar, M.F.S.F.de Moura, Mode II fracture toughness of carbon–epoxy bonded joints with femtosecond laser treated surfaces, International Journal of Mechanical Sciences , vol. 148, 2018, Pages 707-713

https://doi.org/10.1016/j.ijmecsci.2018.09.029

14. P.N.B. Reis, R.A.M. Santos, F.G.A. Silva, M.F S.F. de Moura, Influence of Hole Distance on Low Velocity Impact Damage, Fibers and Polymers, 2018,Vol. 19, Issue 12, Pages 2574–2580

https://doi.org/10.1007/s12221-018-8376-8

Journal impact factor: 1.439; MATERIALS SCIENCE Rank: 5/24 Quartil 1 15. J.P.Reis, M.F.S.F.de Moura, R.D.F. Moreira, F.G.A.Silva, “Pure mode I and II

interlaminar fracture characterization of carbon-fibre reinforced polyamide composite”, Composites Part B, Vol. 169, (2019), Pages 126-132

https://doi.org/10.1016/j.compositesb.2019.03.069

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interlaminar fracture characterization of carbon-fibre reinforced polyamide

composite using the Single-Leg Bending test”, Materials Today Communications, Vol. 19, 2019, Pages 476-481

https://doi.org/10.1016/j.mtcomm.2019.05.006

Journal impact factor: 1.859; MATERIALS SCIENCE Rank: 179/293 Quartil 3 (2018)

17. R.D.F. Moreira, M.F.S.F. de Moura, F.G.A. Silva, F.M.G. Ramírez, J.S.Rodrigues, “Mixed-mode I+II fracture characterisation of composite bonded joints”, Journal of Adhesion Science and Technology (2019)

https://doi.org/10.1080/01694243.2019.1708645

Journal impact factor: 1.210; MECHANICS Rank: 94/138 Quartil 3 (2018) 18. R.D.F. Moreira, M.F.S.F. de Moura, F.G.A. Silva, J.P. Reis, “High-cycle fatigue

analysis of adhesively bonded composite scarf repairs”, Composites Part B 190 (2020) 107900

https://doi.org/10.1016/j.compositesb.2020.107900

Journal impact factor: 6.864; MATERIALS SCIENCE Rank: 1/25 Quartil 1 (2018) 19. R.D.F. Moreira, M.F.S.F. de Moura, F.G.A. Silva, F.M.G. Ramírez, F.D.R. Silva

“Numerical comparison of several composite bonded repairs under fatigue loading”, Composite Structures 243 (2020) 112250

https://doi.org/10.1016/j.compstruct.2020.112250

Journal impact factor: 4.829; MATERIALS SCIENCE Rank: 6/25 Quartil 1 (2018) 20. R.D.F. Moreira, M.F.S.F. de Moura, F.G.A. Silva, “A novel strategy to obtain the

fracture envelop under mixed-mode I+II loading of composite bonded joints”, Engineering Fracture Mechanics 232 (2020) 107032

https://doi.org/10.1016/j.engfracmech.2020.107032

Journal impact factor: 2.908; MECHANICS Rank: 27/134 Quartil 1 (2018)

21. F.M.G. Ramírez, M.F.S.F. de Moura, R.D.F. Moreira, F.G.A. Silva, “A Review on the Environmental Degradation Effects on Fatigue Behaviour of Adhesive Bonded Joints”, Fatigue Fract Eng Mater Struct. 2020;1–20

https://doi.org/10.1111/ffe.13239

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“I feel a thousand capacities spring up in me” Virginia Woolf, The Waves

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1 Introduction I - 1

1.1 Functions of bone tissue I - 1

1.2 Material I - 3

1.3 Objectives I - 8

1.4 Work development I - 10

2 Literature review II - 1

2.1 Fracture of cortical bone tissue II - 1

2.2 Fatigue/fracture of cortical bone tissue II - 18

2.3 Summary II - 28

3 Mixed-mode I+II cohesive zone models III - 1

3.1 Quasi-static mixed-mode I+II cohesive zone model III - 2

3.2 Fatigue mixed-mode I+II cohesive zone model III - 8

3.3 Summary III - 12

4 Numerical analysis of fracture tests IV - 1

4.1 Mode I loading IV - 1

4.1.1 Numerical analysis of the SENB test IV - 2

4.1.2 Numerical analysis of the CT test IV - 5

4.1.3 Numerical analysis of the DCB test IV - 8

4.2 Mode II loading IV - 12

4.2.1 Numerical analysis of the CS test IV - 12

4.2.2 Numerical analysis of the AFPB test IV - 16

4.2.3 Numerical analysis of the ENF test IV - 19

4.3. Mixed mode I+II loading IV – 22

4.3.1 Numerical analysis of the SLB test IV – 23

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5 Experimental work V - 1

5.1 Specimens manufacturing V - 1

5.1.1 Material source V - 2

5.1.2 Milling and cutting operations V - 3

5.2 Design and construction of a new fatigue testing machine V - 14

5.2.1 Design, concept and structure V - 14

5.2.2 Electric and control components V - 16

5.2.3 Upgrades V - 23

5.3 Experimental tests V - 28

5.3.1 DCB tests for pure mode I loading V - 29

5.3.2 ENF tests for pure mode II loading V - 33

5.3.3 SLB tests for mixed mode I+II loading V - 36

6 Results and discussion VI - 1

6.1 Bovine bone VI - 2

6.1.1 Double Cantilever Beam Tests (DCB) VI - 2

6.1.2 End-Notched Flexure tests (ENF) VI - 13

6.1.3 Single-Leg Bending tests (SLB) VI - 22

6.2 Human bone VI - 29

6.2.1 Double Cantilever Beam Tests (DCB) VI - 29

6.2.2 End-Notched Flexure tests (ENF) VI - 32

6.2.3 Single-Leg Bending tests (SLB) VI - 36

6.3. Summary VI - 39

7 Summary, conclusions and perspectives of future work VII - 1

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Figure 1.1. a) Cortical bone and trabecular bone tissues; b) histological section of a trabecular bone filled with marrow.

I-4 Figure 1.2. Representative scheme of the histological constitution of a compact bone

tissue zone.

I-5 Figure. 1.3. a) Crack was deflected by the cement line and propagated around the

osteon; b) crack went through the osteon.

I-6 Figure. 1.4. Representative scheme of the bone self-repair cycle observed in bone

tissue.

I-8 Figure. 2.1. Fracture tests frequently employed to fracture characterization of cortical

bone : Compact tension; b) layer compact sandwich; c) Single-edge-notched beam; d) Chevron-notched beam; e) Double cantilever beam.

II-2

Figure 2.2. Crack propagation revealing extensive fibre-bridging. II-6 Figure 2.3. Direct method: a) GI versus crack opening displacement; b) cohesive

laws, experimental and adjusted; c) load-displacement curves; d) R-curves.

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Figure 2.4. Representation of the three mostly used tests for mode II loading:a) Compact shear (CS); b) End loaded split (ELS); c) End-notched flexure (ENF).

II-8

Figure 2.5. a) The ELS test; b) specimen detail showing the relative shear displacement at the crack notch.

II-8 Figure 2.6. Data from ENF test in bovine cortical bone: a) Load-displacement

curves; b) R-curves.

II-9 Figure 2.7. ENF test in human cortical bone: a) Load-displacement curves; b)

R-curves.

II-10 Figure 2.8. Photography of crack growth under mode II loading. II-11 Figure 2.9. Numerical and experimental curves fit from ENF tests in bovine cortical

bone: a) load-displacement; b) R-curves.

II-12 Figure 2.10. Representation of two proposed tests for mixed-mode I+II loading: a)

single-edge notched asymmetric four-point bending (SEN-AFPB); b) Single Leg Bending (SLB).

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Figure 2.11. Fracture envelope for bovine cortical bone. II-14 Figure 2.12. a) Testing setup of the SLB test; b) detail showing the crack tip under

mixed-mode I+II loading.

II-14 Figure 2.13. a) Numerical and experimental R-curves; b) Representation of the

experimental SLB tests, the power law (=0.8) and the linear fracture criteria (LFC) in the GI-GII space.

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Figure 2.16. Microscope image of crack bridging during cyclically loaded cantilever-beam geometry.

II-18 Figure 2.17. a) Scheme of a human tibia with representation of the orientation and

origin of the CT specimens; b) geometry of obtained CT specimens.

II-19 Figure 2.18. Scanning electron micrographs illustrating crack bridging. For the

transverse orientation, (a, b) evidence of un-cracked-ligament bridging (indicated by black arrows), and (c) possible collagen fibril-based bridging (indicated by black arrows). For the longitudinal orientation, (d) evidence of un-cracked-ligament bridging (indicated by black arrow). The white arrows in (a)–(d) indicate the direction of nominal crack growth.

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Figure 2.19. Typical crack density behaviour in function of relative cycle number for cortical bone tissue.

II-22 Figure 2.20. Trace from crack path growth deflected around osteon. II-23 Figure 2.21. Typical fatigue behaviour. Bi-logarithmic plot of crack growth rate as

function of normalized strain energy release rate.

II-27 Figure 3.1. Mixed-mode I+II cohesive law with trapezoidal softening relationship

between stresses and relative displacements.

III-3 Figure 3.2. Bilinear cohesive law for mixed-mode I+II loading used in high-cycle

fatigue analysis.

III-8 Figure 4.1. Geometry of the SENB specimen analysed. IV-2 Figure 4.2. C=f(a) relationship adjusted by polynomial with different degrees; a)

third; b) fourth; c) sixth; d) the corresponding GI=f(a) curves.

IV-3 Figure 4.3. Evolution of; (a-c) normal stresses during crack growth; d) FPZ length as

function of crack length in the SENB test.

IV-5 Figure 4.4. Geometry of the CT specimen analysed. IV-6 Figure 4.5. C=f(a) relationship adjusted by polynomial with different degrees; a)

third; b) fourth; c) sixth; d) the corresponding GI=f(a) curves.

function of crack length.

IV-8 Figure 4.7. Geometry of the DCB specimen analysed. IV-9 Figure 4.8. C=f(a) relationship adjusted by polynomial with different degrees; a)

third; b) fourth; c) sixth; d) the corresponding GI=f(a) curves.

function of crack length in the DCB test.

IV-11 Figure 4.10. a) Geometry of the CS specimen analysed and b) loading conditions. IV-13 Figure 4.11. Mesh used in the finite element analysis for the CS test. IV-13

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Figure 4.13. Normalized R-curve for the CST test. IV-15 Figure 4.14. Schematic representation of the AFPB test. IV-16 Figure 4.15. Mesh from numerical simulation of the AFPB test under mode II loading,

showing crack propagation.

IV-16 Figure 4.16. Stress profiles along the horizontal crack path ahead of the crack-tip just

before; (a) damage onset and (b) crack starting advance.

IV-17 Figure 4.17. Normalized R-curve, function of longitudinal crack extension, obtained

for AFPB test.

IV-18 Figure 4.18. Schematic representation of the ENF test. IV-19 Figure 4.19. C=f(a) relationship adjusted by polynomial with different degrees; a)

third; b) fourth; c) sixth; d) the corresponding GII=f(a) curves.

IV-20 Figure 4.20. Evolution of; (a-c) shear stresses during crack growth; d) FPZ length as

function of crack length in the ENF test.

IV-21 Figure 4.21. Schematic representation of the SLB test. IV-22 Figure 4.22. C=f(a) relationship adjusted by polynomial with different degrees; a)

third; b) fourth; c) sixth; d) the corresponding GT=f(a) curves.

IV-23 Figure 4.23. Evolution of; (a-c) von Mises stresses during crack growth. IV-24 Figure 4.24. Evolution of a) FPZ length as function of crack length in the SLB test; b)

fracture envelope for mixed-mode I+II considering the linear energetic criterion.

IV-24

Figure 5.1. Detail of the bovine bone showing the small part resulting from the first cut.

V-8 Figure 5.2. Production reference surface in the small piece: a) original geometry; b)

polishing with water sandpaper at 3600 rpm; c) quality of the obtained surface; d) final aspect after polishing operations.

V-10

Figure 5.3. Milling operations: a) milling with water flow; b) final aspect after milling until intended thickness.

V-10 Figure 5.4. a) Cutting the unwanted edges; b) block with a constant thickness

between two parallel faces.

V-11 Figure 5.5. a) Positioning of bone for cutting edges to obtain reference surfaces; b)

alignment between bone and cutting disc.

V-12 Figure 5.6. Measurement of the distance between the steel reference plate and cutting

disc.

V-12 Figure 5.7. Positioning of the bone between steel plates for cutting process. V-13 Figure 5.8. Aspect of several specimens after the performed precision cuts. V-13

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XX right extremity.

Figure 5.10. Assembly for production the longitudinal groove with references faces and cutting disc with V-shape: a) concept; b) reality.

V-15 Figure 5.11. Diamond blade (0.3 mm thickness) aligned with the groove. V-16 Figure 5.12. a) Reference lines for specimen alignment; b) detail of the cut beginning;

c) final position to ensure a correct notch.

V-16

Figure 5.13. Premature crack propagation. V-17

Figure 5.14. a) setup for the introduction of the pre-crack; b) final aspect of the notch and the pre-crack.

V-17 Figure 5.15. Specimens obtained for one of the fracture testing campaigns. V-18 Figure 5.16. Specimen geometry with nominal dimensions in millimetres. V-18 Figure 5.17. a) Setup to drill aligned holes in the DCB specimens; b) detail of

references surfaces to accommodate the specimen.

V-19 Figure 5.18. Specimen for the mixed mode I+II fracture tests (SLB). V-19 Figure 5.19. a) Preliminary draft drawing; b) photography of developed equipment. V-22 Figure 5.20. Threaded spindle driven table actuated by a stepping motor from IGUS®. V-23 Figure 5.21. a) Micro-step driver for controlling stepper motor movements; b)

DRFArduino UNO model printed circuit board; c) digital to analogical amplifier and converter HX711.

V-24

Figure 5.22. Overview of fatigue test machine with hydration system. V-26 Figure 5.23. a) Step pump that pushes the syringe, providing a controlled serum flow;

b) protection system of the components and accessories of the test machine and its serum tank.

V-27

Figure 5.24. PLA cover to protect metallic parts from chemical attack and to support the specimen.

V-27 Figure 5.25. Design model and photography of the PNP transistor based security

system.

V-28 Figure 5.26. Parts of safety system with a square section holes, for making the system

more stable without rotation; a) left support; b) length limiter; c) right support for sensor PNP.

V-29

Figure 5.27. Photography of the right support with PNP transistor based security system.

V-30 Figure 5.28. a) Wheatstone Amplifier Shield (RB-Onl-38); b) ADC Data Acquisition

Shield (ARD-LTC1867).

V-31 Figure 5.29. Magnetostrictive linear position sensor. V-31

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Figure 5.31. Test fixtures for flexural tests with reference points to facilitate alignment.

V-33 Figure 5.32. Pin support for traction tests with: a) small loads (bone); b) other

materials.

V-33 Figure 5.33. Top view of the developed new equipment. V-34 Figure 5.34. Schematic representation: a) bone, showing the origin and orientation of

the specimen; b) setup for DCB test.

V-35 Figure 5.35. DCB test in the new machine revealing details of hydration and load

application.

V-36 Figure 5.36. DCB specimen showing a crack propagating in the middle plane of the

specimen during a fatigue test.

V-37 Figure 5.37. Schematic representation: a) bone, showing the origin and orientation of

the specimen; b) assembly of the ENF test.

V-39 Figure 5.38. ENF specimen in the test machine: a) Alignment: b) testing. V-40 Figure 5.39. Schematic representation: a) bone, showing the origin and orientation of

the specimen; b) assembly of the SLB test

V-42 Figure 5.40. SLB specimen with the addition of a 0.3 mm thick spacer and the cut arm

remains, glued with tape to loaded arm.

V-44 Figure 5.41. Fatigue/fracture SLB test: a) crack propagation at the specimen

mid-plane; b) typical final failure characterized by crack propagation in the mid-plane until arm rupture occurs.

V-44

Figure 6.1 Deformation field obtained by DIC analysis. VI-3 Figure 6.2 Results of a DCB quasi-static test of bovine cortical bone: a)

load-displacement curve; b) R-curve.

VI-4 Figure 6.3 Detail of crack propagation (top view) during a DCB fatigue test

revealing the presence of the longitudinal grooves and irrigation in the course of the test.

VI-5

Figure 6.4 Typical DCB high-cycle fatigue test of bovine cortical bone: a) load versus number of cycles; b) displacements for maximum and minimum applied loads,  (Pmax) and  (Pmin).

VI-6

Figure 6.5 a) Compliance and b) equivalent crack length, of bovine cortical bone both represented as function of the number of cycles.

VI-7 Figure 6.6 a) Evolution of the variation of strain energy release rate versus number

of cycles; b) fatigue damage growth (dAe/dN - mm2/cycle) versus the ratio GI/GIc in a bi-logarithmic representation,from bovine cortical bone test.

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bi-logarithmic representation for fatigue/fracture behaviour of this set of bovine bone specimens.

Figure 6.8 Mesh and boundary conditions used in the DCB simulations. VI-10 Figure 6.9 Magnification (20x) of fatigue fracture surface from bovine bone under

mode I loading.

VI-13 Figure 6.10 Visualization of sequential unwanted brittle fracture observed in ENF

fatigue tests without hydration.

VI-14 Figure 6.11 Results of an ENF quasi-static test of bovine cortical bone: a)

VI-15 Figure 6.12 ENF specimen with a full-developed crack. VI-16 Figure 6.13 Typical ENF high-cycle fatigue test of bovine cortical bone: a) loads

versus number of cycles; b) displacements for maximum and minimum load applied,  (Pmax), (Pmin).

VI-17

Figure 6.14 a) Compliance and b) equivalent crack length, of bovine cortical bone both in function of the number of cycles.

VI-18 Figure 6.15 a) Evolution of the variation of strain energy release rate versus number

of cycles; b) fatigue damage growth (dAe/dN - mm2/cycle) versus the ratio GII/GIIc in a bi-logarithmic representation, from bovine cortical bone test.

VI-18

Figure 6.16 Normalized compliance versus number of cycles of valid results; b) fatigue damage growth (dAe/dN - mm

2

/cycle) versus the ratio GI/GIc in a bi-logarithmic representation for fatigue/fracture behaviour of this set of bovine bone.

VI-18

Figure 6.17 Mesh, boundary conditions and load application considered in the ENF simulations.

VI-20 Figure 6.18 Magnification (20x) of fatigue fracture surface from bovine bone under

mixed-mode I+II loading.

VI-21 Figure 6.19 Results of a SLB quasi-static test of bovine cortical bone; a)

VI-23 Figure 6.20 Fracture envelope for this set of specimens of bovine cortical bone. VI-23 Figure 6.21 SLB specimen after testing showing some crack extend (a) under

self-similar condition.

VI-24 Figure 6.22 Typical SLB high-cycle fatigue test of bovine cortical bone: a) loads

versus number of cycles; b) displacements for maximum and minimum load applied,  (Pmax) and  (Pmin).

VI-25

Figure 6.23 SLB specimen: a) compliance and b) equivalent crack length, of bovine cortical bone both in function of the number of cycles.

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XXIII

ratio GT/GTc in a bi-logarithmic representation, from bovine cortical bone test.

Figure 6.25 a) Normalized compliance versus number of cycles for the valid tests; b) fatigue damage growth (dAe/dN - mm2/cycle) as function of GT/GTc in a bi-logarithmic representation of this set of bovine bone specimens.

VI-26

Figure 6.26 Plot of C1m and C2m as function of the mode-mixity and of the Paris law parameters ensuing from the SLB test (Figure 6.25b).

VI-27 Figure 6.27 Mesh and boundary conditions used in the SLB simulations. VI-28 Figure 6.28 Magnification photography (20x) of fatigue/fracture surface of bovine

bone under mixed-mode I+II loading.

VI-28 Figure 6.29 a) Compliance in function of the number of cycles; b) fatigue damage

growth (dAe/dN - mm2/cycle) versus the ratio GI/GIc in a bi-logarithmic representation of human bone specimen.

VI-29

Figure 6.30 Results of a DCB quasi-static test in human cortical bone; a) load-displacement curve; b) R-curve.

VI-30 Figure 6.31 Normalized compliance versos number of cycles of valid results; b)

fatigue damage growth (dAe/dN - mm 2

/cycle) versus the ratio GI/GIc in a bi-logarithmic representation for fatigue/fracture behaviour of this set of human bone.

VI-32

Figure 6.32 a) Compliance in function of the number of cycles; b) fatigue damage growth (dAe/dN - mm2/cycle) versus the ratio GII/GIIc in a bi-logarithmic representation of human bone specimen.

VI-33

Figure 6.33 Results of an ENF human bone in quasi-static test; a) load-displacement curve; b) R-curve.

VI-33 Figure 6.34 a) Normalized compliance versus number of cycles of valid results; b)

fatigue damage growth (dAe/dN - mm2/cycle) versus the ration GII/GIIc in a bi-logarithmic representation for fatigue/fracture behaviour of this set of human bone.

VI-35

Figure 6.35 Results of a SLB human bone quasi-static test; a) load-displacement curve; b) R-curve.

VI-36 Figure 6.36 Fracture envelope for this set of specimens of human cortical bone. VI-36 Figure 6.37 SLB specimen: a) compliance in function of the number of cycles; b)

fatigue damage growth (dAe/dN - mm2/cycle) versus the ratio GT/GTc in a bi-logarithmic representation of human bone specimen.

VI-37

Figure 6.38 a) Normalized compliance versus number of cycles for the valid tests; b) fatigue damage growth (dAe/dN - mm2/cycle) as function of GT/GTc in a bi-logarithmic representation of valid results for this set of human bone.

VI-38

Figure 6.39 Plot of C1m and C2m as function of the mode-mixity and of the Paris law parameters ensuing from the SLB test (Figure 6.38b).

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XXIV

Table 6.1. Resume of bovine DCB specimen dimensions, properties and fatigue lives. VI-9 Table 6.2. Elastic properties used in the numerical models for bovine and human bone VI-11 Table 6.3. Fracture properties under mode I loading for each bovine specimen. VI-12 Table 6.4. Resume of bovine ENF specimen dimensions, properties and fatigue lives. VI-19 Table 6.5. Fracture properties under mode II loading for each bovine specimen. VI-20 Table 6.6. Resume of bovine SLB specimen dimensions, properties and fatigue lives. VI-27 Table 6.7. Resume of human bone specimens origin. VI-29 Table 6.8. Resume of DCB specimen dimensions, properties and fatigue lives for

human cortical bone tissue.

VI-31 Table 6.9. Fracture properties under mode I loading for each specimen of human

cortical bone tissue.

VI-31 Table 6.10. Resume of ENF specimen dimensions, properties and fatigue lifes for

human bone.

VI-34 Table 6.11. Fracture properties under mode II loading for each specimen of human

bone.

VI-34 Table 6.12. Resume of SLB specimen dimensions, properties and fatigue lives for

human cortical bone tissue.

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XXV

Latin

a (mm) crack length

ae (mm) equivalent crack length

a0 (mm) initial crack length

Ap,k (mm2) local damaged area

At,k (mm2) area corresponding to each integration point k

B (mm) specimen width

b (mm) width of the resistant section

c (mm) distance in the AFPB test

C (mm/N) specimen compliance

Ci  adjustment parameters for Paris law (i=1, 2)

Cij  fatigue parameters of the modified Paris law (i=1, 2) under

pure modes loading (j=I, II)

Cim,k  fatigue parameters (i=1, 2) for the current mode ratio in

integration point k

D  diagonal matrix containing the damage parameter

d (mm) distance in the AFPB test

df  fatigue damage parameter

df  variation of the fatigue damage parameter

ds  static damage parameter

N  numerical cycle jump

E MPa matrix elastic modulus

Ei MPa elastic modulus in direction i (i=L, R, T)

GB (N/mm) mixed-mode strain energy release rate at point B of the cohesive law

Gc (N/mm) critical fracture energy

j

G_d (N/mm) total energy dissipated an increment j

j

G

 (N/mm) increase of energy dissipation in each mode i (i=I, II) at increment j

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XXVI

Gi (N/mm) strain energy release rate under mode i (i=I, II) loading Gic (N/mm) critical fracture energy under mode i (i=I, II) loading

GLT (N/mm) shear modulus in the LT plane

GT (N/mm) total mixed-mode I+II strain energy release rate

GTc (N/mm) critical value for total mixed-mode I+II strain energy release rate

j

G_{T c,}_k (N/mm) fracture energy in mixed-mode I+II for integration point k at increment j

GTd (N/mm) energy dissipated in mixed-mode I+II

j k

GT d, (N/mm) energy dissipated in mixed-mode I+II for integration point k

at increment j

min T , k

G (N/mm) total mixed-mode I+II strain energy release rate at Pmin for a integration point k

max T , k

G (N/mm) total mixed-mode I+II strain energy release rate at Pmax for a integration point k

k

GT ,

 (N/mm) variation of total strain energy release rate in a cycle

j k

G_{T ,}_max, (N/mm) maximum value strain energy release rate in a given integration point k for each increment j under mixed-mode I+II loading h (mm) specimen half-height I  identity matrix k0 (N/mm3) initial stiffness L (mm) specimen half-length lc (mm) characteristic length N  number of cycles

nFPZ  number of the integration points undertaking softening in the fracture process zone

P (N) applied load

Pmin (N) minimum applied load during the fatigue test

Pmax (N) maximum applied load during the fatigue test

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XXVII

integration scheme

k  integration point undergoing fatigue damage

u (mm) crack tip shear displacement

w (mm) crack opening displacement

y (mm) distance in the AFPB test

x (mm) distance in the AFPB test

Greek

  stress ratio

  displacement ratio

  exponent value from the power law energy based criterion

 (mm) applied displacement

i (mm) mode i (i=I, II) components of the mixed-mode displacement

i (mm) relative displacements for damage initiation under pure mode i (i=I, II) loading

m (mm) equivalent relative displacement under mixed-mode I+II

loading

1m (mm) mixed-mode equivalent relative displacement at damage onset

1m,i (mm) component of relative displacements (i=I, II) at damage initiation under mixed-mode I+II loading

i m (mm) equivalent relative displacement at critical points (i=1, 2) of the cohesive law

j

δm (mm) displacements for damage increment

um (mm) ultimate relative displacement under mixed-mode I+II loading

j k

δ₁_m_, (mm) damage onset relative displacement under mixed-mode I+II loading in a given integration point k for the increment j

j k

δ_um_, (mm) ultimate relative displacement under mixed-mode I+II loading in a given integration point k for the increment j

i (mm) relative displacements for damage initiation under pure mode i (i=I, II)

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XXVIII

increment j

j k i

δ_, (mm) relative displacement under pure-mode i (i=I, II) for

increment j

i MPa stress component i (i=I, II) of the mixed-mode I+II loading

1,i  MPa local strength under pure mode loading i (i=I, II)

m MPa equivalent stress under mixed-mode I+II loading

j m

 MPa local stress under mixed-mode I+II for increment j

1m,i MPa stress component i (i=I, II) of the mixed-mode I+II at damage onset

um  MPa maximum equivalent local stress under mixed-mode I+II

j k , um 

MPa maximum equivalent local stress under mixed-mode I+II at point k for increment j

u,i  MPa maximum local strength under pure-mode i (i=I, II)

j i

 MPa stress component i (i=I, II) at increment j

1



j

i MPa stress component i (i=I, II) at damage increment j-1

Acronyms

ADC analog-to-digital converter

CBBM compliance based beam method

CDM continuum damage model

CNB chevron-notched beam

COD crack open displacement

CS compact shear

CT compact tension

CZM cohesive zone model

DCB double cantilever beam

DIC digital image correlation

ELS end-loaded split

ENF end-notched flexure

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XXIX

LEFM linear elastic fracture mechanics

LFC linear fracture criteria

MMB mixed-mode bending

PLA polylactic acid or polylactide

SCS single-layer compact sandwich

SENB single-edge-notched beam

SEN-AFPB single-edge notched asymmetric four-point bending test

AFPB asymmetric four-point bending test

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Chapter 1 - Introduction Page I - 1

Chapter 1 Introduction

1.1 Functions of bone tissue

Bones, as well as the complete skeletal system, play very important roles for the functioning of the human body. The primary function of the skeletal system is body support. The perfect symbiosis between bones and muscles allows for balanced movements, allowing walking, running and jumping in perfect harmony. Protection is another very important function of some bones. For example, ribs and skull bones provide protection for vital organs like lungs, heart and brain from impact loading. The third relevant function of bone tissue is its capacity of storing some minerals with fundamental importance to the human body, such as phosphorus and calcium. When the need for these minerals occurs, bones release them to maintain mineral balance [1].

Bone tissue is a material that, during its life, is continually changing to preserve the mechanical properties. It is a remarkable biological material, with the unique ability to build very resilient structures that reshape and repair themselves. It has excellent regenerative capacity when lesions occur, through a process of replacement or reconstruction of the affected areas. Bone tissue also shows a remarkable aptitude to adapt to complex load distributions, which generally reduces the occurrence of fractures [1]. The self-healing ability of living bone tissue provides repair of small fissures (typically around 0.1 mm) before spreading. Nevertheless, bone fractures are quite common and can be grouped into three main categories depending on its origin:

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traumatic, pathological and fatigue [2]. Therefore, predicting the risk of bone fracture is a relevant research topic given its socioeconomic impact. In particular, the prediction of age-related changes in fracture behaviour is of great importance, given the increase of longevity of the population. In fact, the number of elderly population is rising and, at present date, it is considered the fastest growing age group in the all world [3]. Most of the studies are focused on age-related changes due to the inherent increase in bone fragility. In addition, bone tissue is not immune to problems. In fact, cortical bone tissue is a living material that suffers profound alterations in its chemical composition and structure as a function of the environment, age, nutrition, drugs and other factors. Various diseases and pathologies are associated with bone tissue. Aging is often associated with diseases with osteoporosis, which is characterized by the loss of bone density that occurs gradually. This phenomenon leads to a weakening of the bone tissue, making it favourable to the appearance of fractures. Beyond bone diseases, fracture in cortical bone tissue is very likely to occur due to accidental causes, fatigue loading as a result of falls, jumps, walking or running. The occurrence of fractures among young people is usually associated with high-competition of athletes, dancers and military personnel who are subjected to repeatedly impact, or involved in intense physical activities. These fractures, clinically designed by fatigue fractures, have an incidence rate between 0.7% to 20% for all sports injuries [4], which justifies the development of studies on this topic. Another example of frequent bone fracture occurs in hip, constituting a serious and costly public health problem. Haentjens et al. [5] reported an excess of mortality following a hip surgery fracture due to complications associated with bone healing. The authors stated that there is an increase of mortality risk up to eight times following a hip fracture in elderly people in the three months subsequent to the event. Hip fractures are very common in the elderly people because of their reduced

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capacity to repair damage/micro-cracks. Approximately 1.5 million hip fractures happen every year worldwide, and the statistical studies points to an estimate of 2.6 million by 2025, with a grownup until 4.5 million by 2050 [5-7]. Therefore, it is expected an increase of costs in public health and a reduced quality of life.

Bone fractures are analysed by the scientific community from the point of view of those ones that have been induced by single loads in fragile bones. However, in view of the inability or individual’s deficient self-repair capability, small cracks and decreased elasticity, it is believed that fractures may be the result of small daily loads acting on fragile bones [5]. For bone tissue, the declining fatigue resistance and fracture toughness appears to be a complicated function of alterations in bone mass, rate of remodelling/self-repair and potential changes in the collagen that is responsible for bone flexibility.

1.2 Material

Bone tissue is constituted by a mineral phase (hydroxyapatite) and an organic phase (collagen). The unique properties of bone tissue are due to the combination of these two phases as they perfectly associate the rigidity of the mineral phase with the elasticity of collagen. Because of these two parts, bone tissue becomes resistant to mechanical stresses, provides support and protection functions (mineral phase) and presents a certain flexibility (organic phase). As age increases, the modification of these two phases leads to changes in properties, like the reduction of elasticity and the increase of brittleness.

Considering the structural organization, bone tissue can be classified as trabecular or cortical (Figure 1.1a) [1]. Both have the same constituent elements differing in their structural organization and functionality. Effectively, cortical bone tissue has a compact

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and organized structure while the trabecular one reveals a spongy structure in three-dimensional network, arranged according to the directions of mechanical loads and filled by bone marrow. In Figure 1.1b, the histological section shows a trabecular bone filled with marrow.

a) b)

Figure 1.1. a) Cortical bone and trabecular bone tissues; b) histological section of a trabecular bone filled with marrow [1].

Cortical bone tissue is formed by cylindrical structures (called Harversian systems - Figure 1.2), with an average diameter of 200 to 250 m and about 1 cm in length, oriented parallel to the longitudinal axis (L) of the bone [8]. These structures are surrounded by a mineralized matrix with a thickness of 1 to 2 m. This structural organization gives the material orthotropic properties with three preferred directions: L-longitudinal; R-radial; T-transverse. Due to different structural organization, cortical bone tissue is less porous than the spongy one, being highly irrigated by many blood vessels and filled with nervous ramifications. Consequently, the modulus of elasticity and the mechanical strength of the cortical bone tissue are substantially greater than that of the trabecular bone, which is fundamental owing to supporting and protective functions. Consequently, the majority of the studies are dedicated to analyse cortical bone tissue.

Cortical bone

Trabecular bone

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Figure 1.2. Representative scheme of the histological constitution of a compact bone tissue zone (adapted from [1]).

Some of these studies address fracture toughness of the different microstructures that constitute the cortical bone tissue. Various bone fracture toughness measurements have been obtained and attributed to the different microstructures and their constituents that define the existent orientations [7, 9, 10, 11-13]. The analyses of the crack path propagation around the osteons, in cement lines and sometimes through the osteons are widely observed with microscopy. Figure 1.3a presents a crack that was deflected by the

Periostium External circumferential lamellae Volkmann’s canal Harversian canal Interstitial Lamellae

Osteon (circumferential lamellae from a Harversian system) Orientation of collagen fibers

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cement line (outer region of an osteon) and find a path around the osteon, while Figure 1.3b shows an example of a crack propagating through an osteon [7]. Several studies focus their attention on crack micro-scale behaviour related to same aspects like, osteocyte lacunar density, mineralization of collagen fibres or accumulation of micro-cracks [3, 4]. These studies analysed the influence of a micro component from bone hierarchal organization in crack deviation or propagation, considering several aspects as aging, mineralization ratio, bone degradation and osteoporosis [14, 15].

Figure. 1.3 a) Crack was deflected by the cement line and propagated around the osteon; b) crack went through the osteon [7].

A wide choice in the origin of bones analysed in the various studies presented in the literature reveals that bones from different species are similar to the human bones. The large majority of studies are performed on cadaver bones. However, few works on live animals focus not only on fracture toughness but also on regeneration of applied damaged in bone tissue during the test (Figure 1.4). Some studies have focused on obtaining methods to determine bone fracture toughness using small and/or large animals. Species such as murine [16-18], rabbits [19], canine [17], pigs [9], monkeys and baboons [20], horses [21], bovine [10, 21, 22], manatee rib [22], buffalo [23] and elephant [24] have been considered. The characterization of dentin [24, 25] is also

Cement

line

Through

_{the osteon}

(b)

(a)

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widely approached given its socio-economic impact for reasons related to nutrition as well as aesthetic aspects.

Figure 1.4 shows a schematic representation of the self-repair cycle, that has four different stages. Considering that the cycle starts after a damage occurrence, the first stage (during 7-10 days) is the Resorption and consists of an erosion task performed by the osteoclasts that were previously activated. In this cycle, bone mineral and matrix is removed to create a cavity. In the second stage, Reversal, the surface cavity is prepared to receive the new osteoblasts to begin forming new bone tissue, which is accomplished by mononuclear cells. Next stage is Formation (duration of 3 months), which consists of osteoblasts arriving layer-by-layer and starting to synthesize an organic matrix to fulfil the cavity with new bone. Resting is the last stage that involves covering the bone surface with flattened lining cells, thus completing the repair sequence [26].

Figure. 1.4. Representative scheme of the bone self-repair cycle observed in bone tissue. [26].

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1.3 Objectives

Nowadays, fatigue and fracture behaviour of hard tissues are topics of considerable interest. This special group of organic materials comprises the highly mineralized and load-bearing tissues of the human body, and includes bone, cementum, dentin and enamel. An understanding of their fatigue behaviour, the influence of loading conditions and physiological factors (e.g. aging and disease) on the mechanisms of degradation are essential for achieving lifelong health. However, there is much more to this topic than the immediate medical issues. There are many challenges to characterize the fatigue behaviour of hard tissues, much of which are attributed to size constraints and the complexity of their microstructure [8, 25-27]. The knowledge about initiation and crack growth in bone tissue and the influence of those aspects is still in the early stages, since fundamental information remains unknown. The complexity of bone structure and the advance of design solutions for load support and fracture resistance are a tremendous source of inspiration for researchers and engineers [25]. As discussed above, elderly people is the fastest growing age group usually associated with diseases like osteoporosis and hip fractures, which are two major events that represent loss of quality live and costly public health problem. A loss of bone quality is also associated with lack of ability of self-repair. In fact, without repair, micro-cracks originated from daily loads tend to grownup and became a bone fatigue/fracture problem. Hence, fatigue/fracture is prone to occur when self-repair mechanisms (Figure 1.4) [26] do not have time to repair micro-cracks originated from small loads that are repeatedly applied.

It is then crucial to develop methodologies that determine fracture properties to evaluate the healthy condition of bone. In this context, the mechanical, fracture and

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fatigue/fracture characterizations of cortical bone tissue are relevant aspects that have been increasing the interest of the scientific community. In the actual research team, new methods have been successfully applied for quasi-static fracture characterization of bovine and human bone under pure mode I, pure mode II and mixed mode I+II loading [28-41]. These new methodologies take into account that bone has non-linear fracture mechanics behaviour resulting from the complex and hierarchal structure that are responsible for development of a significant fracture process zone (FPZ). These approaches walk according with several others that support the conviction that bone behaviour needs to be analysed considering non-linear fracture mechanics [42-45]. Another advantage of these methodologies is the employment of simple tests, the Double Cantilever Beam (DCB) for pure mode I, the End-Notched Flexure (ENF) for pure mode II and the Single-Leg Bending (SLB) for mixed mode I+II fracture characterization of cortical bone tissue. Since these tests are able to characterize bone fracture behaviour for quasi-static loading, they will be adopted in this work for cortical bone characterization in the context of fatigue/fracture. Additionally, the development of cohesive zone models based on the modified Paris law suitable for the simulation of fatigue/fracture under different loading modes is foreseen.

The main objective of this work is to propose reliable methodologies for the fatigue/fracture characterization of bone, thus allowing systematic studies addressing aspects such as the influence of age, drugs, diseases, trauma and the influence of lack of some vitamins and nutrients. These studies aim to better understand fracture behaviour of bone and contribute to choose the best clinical treatment to recover or minimize the bone quality loss that is frequently the cause of painful fractures.

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1.4 Work development

As a result of the specified objectives, this PhD Thesis is organized as follows:

 In Chapter 2, a detailed literature review focussing on the actual state-of-the-art of the experimental methodologies and numerical approaches addressing fracture and fatigue/fracture characterization of cortical bone tissue is presented;

 In Chapter 3, a comprehensive description of the numerical quasi-static and fatigue cohesive zone models employed along the Thesis for the sake of validation procedures is done;

 Chapter 4 addresses a numerical analysis focussed on the most used fracture tests applied to bone and proposed in literature, aiming to discuss their limitations;

 Chapter 5 describes all the performed experimental work, being organized in three sections describing the essential experimental tasks: Specimens manufacturing; Design and construction of a new testing machine; Experimental tests;

 Chapter 6 presents and discusses the main results of the fracture and fatigue/fracture tests;

 Chapter 7 summarizes the main relevant conclusions of the performed work and lists the perspectives of future work.

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Chapter 2 – Literature review Page II - 1

Chapter 2 Literature review

2.1 Fracture of cortical bone tissue

Fracture of cortical bone tissue has been the object of study by the scientific community, due to the growing interest motivated mainly by aging of the population. In general, bone fractures start from small fissures that propagate under various modes of loading. In this context, the employment of Fracture Mechanics (FM) concepts is adequate [46], as it allows the determination of the material toughness, a property that quantifies the resistance to crack initiation and propagation. Most current applications are limited to the use of Linear Elastic Fracture Mechanics (LEFM). Norman et al. [2] considered the concepts of LEFM and employed the Compact Tension (CT) test (Figure. 2.1a) [37] to compare the fracture toughness of human and bovine bone. Several aspects were analysed to determine their contribution for fracture toughness. The authors admit a mal-function of the LEFM approaches since the value of energy release rate increases with crack propagation and never stabilises. Vashishth et al. [47] executed fracture tests in CT specimens and analysed the microstructural damage near the crack tip. The authors reported that the number of micro-cracks increases linearly with the main crack length. In another work, Vashishth et al. [48] analysed the relation between the velocity of a crack during the test and the new emergent group of micro-cracks in the vicinity of the crack tip. The acceleration and deceleration of the main crack growth varied with micro-cracks development, indicating that main crack only

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develops after the complete development of a zone with micro-cracks. Akkus et al. [49] investigated how multi micro-cracks grow and converge into a main one and their interaction with the different micro-structures that compose the human bone. Two different specimens’ orientations (longitudinal and transverse) were studied with CT specimen considering a middle plane groove to keep propagation at this plane. The authors observed differences between crack behaviour and growth in the two orientations. They also found that toughness in transverse direction is significantly greater than in the longitudinal direction.

Figure. 2.1. Fracture tests frequently employed to fracture characterization of cortical bone: Compact tension; b) Single-layer compact sandwich; c) Single-edge-notched

beam; d) Chevron-notched beam; e) Double cantilever beam [37]. d)

c)

e)

a)

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Fracture of cortical bone tissue is characterized by several mechanisms of damage, such as micro-cracking, fibre bridging and crack deviation [23, 50]. These damage mechanisms give rise to a non-negligible fracture process zone, making the application of LEFM inappropriate. Yang et al. [44] analysed data from CT tests in human humeral cortical bone considering two approaches: the LEFM and a non-linear fracture mechanics approach considering a cohesive zone model. The authors observed that the standard LEFM is unable to mimic the load–displacement curves but the nonlinear model was able to predict several load–displacement curves after calibration with data from one curve. The predictions of the cohesive zone model are in line with previous observations, which estimate that the length of the nonlinear damage zone is in the range between 3 to 10 mm. Taking into account the specimen dimensions, the size of this damage zone is considerably large, leading to the conclusion that LEFM is not an accurate model to be applied for cortical bone tissue. Effectively, the nonlinear cohesive zone model is a more accurate approach to predict bone fracture behaviour. The cohesive law offers a better representation of the failure mechanics than a simpler fracture toughness parameter [44].

The majority of the studies present in literature regarding bone fracture characterization are about mode I loading. In this context, the CT test is widely used for bone fracture characterisation of that loading mode [47-49, 51-57]. This test consists of a small bone specimen with a pre-crack under tension loading which lead to mode I opening displacements at the crack tip (Figure. 2.1a). Considering bone specimens size restrictions a variant of CT test, namely single-layer compact sandwich (SCS) (Figure 2.1b) was used [10, 20, 58]. This test consists of a small piece of the testing material (bone in this case) bonded to two holders that provide the specimen size with dimensions similar to the CT specimen. Two more test configurations have been used:

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the single-edge-notched beam (SENB) (Figure2.1c) [20, 45, 50, 59] and the chevron-notched beam (CNB) (Figure 2.1d) test [22, 60]. Both are based on a three point bending configuration considering a specimen with a small notch at the specimen mid-span. The difference between SENB and CNB tests is the shape of the notch and the resulting cross-section geometry, a square cut and an isosceles triangular cross-section (image underneath Figure 2.1d), respectively. In the following some works employing the referred tests are described.

Norman et al. [51] compared the value of critical strain energy release rate of human cortical bone tissue in longitudinal direction under mode I and mode II loading using the CT and compact shear (CS) tests (Figure 2.4). The authors used bone harvested from cadavers (ranging age: 55 - 89 years old; male and female) and measured the critical strain energy release rate. Since it was not possible to obtain the individual compliance calibration, an average of all donors was employed. The authors point that no significant differences were found for fracture toughness between males and females, and they observed that bone resistance in longitudinal axis diminishes with age. In another work, Norman et al. [52] analysed the density of micro damage in the vicinity of the crack tip. The authors concluded that there is an inverse relationship between fracture toughness and damage density. Phelps et al. [20] evaluated the fracture toughness considering microstructural heterogeneity in baboon bone. The main goal was to determine the differences in proprieties between osteonal and interstitial regions. Analyses of longitudinal and transverse fracture toughness were performed employing two fracture tests, SCS and SENB. Results suggest that longitudinal fracture toughness decreases significantly with age but interstitial micro-hardness increases. For transverse fracture toughness, the increase of micro-hardness in interstitial region does not have any significant effect. Wang et al. [10] determined the fracture toughness of bovine

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bone using a compact sandwich specimen applying LEFM. Considering wet and dry bone at two different crack orientations, the samples came from three different localizations of the specimens source, the inside, the middle and outside (of radial axis). Authors concluded that bone toughness depends on radial specimens’ source and orientation. In another work, Wang et al. [58] studied whether fracture toughness of bone can be predicted from mechanical properties and mineral density of the baboon bone. In conclusion, the authors assumed that baboon bone fracture toughness is correlated with microhardness, but not with other parameters (porosity, yield strength, mineral density, ultimate strength and elastic modulus). Li et al. [45] performed experimental and numerical studies considering the SENB test in bovine cortical bone tissue. The authors concluded that the different axial directions reveal different fracture toughening mechanisms. They also verified that bovine cortical bone develops a non-uniform elastic-plastic fracture process zone. Kruzic et al. [24] studied fracture mechanisms of elephant dentin and identified remarkable toughening characteristics based on the Resistance-curve (R-curve) behaviour. Microscope and X-ray tomography observations revealed crack bridging as the relevant toughening mechanism. Comparison between hydrated and dehydrated dentin shows an important crack blunting responsible for higher fracture resistance in the first one relative to the second case. The authors also admit that constrained micro-cracking leads to toughness increases.

Morais et al. [28] studied two new approaches to determine fracture toughness in hydrated and thermally dehydrated cortical bovine bone under pure mode I loading. The authors considered non-linear fracture mechanics via cohesive zone modelling, and determined the corresponding R-curves. A miniaturized version of the double cantilever beam (DCB) test (Figure 2.1e) was successfully applied. This test allows to apply a data

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reduction scheme based on specimen compliance and equivalent crack length concept to obtain the complete R-curve without monitoring crack evolution during the test. In conclusion, the authors presented the DCB as an effective test for pure mode I fracture characterization of cortical bone tissue. Pereira et al. [31] also used the DCB test to determine the cohesive laws representative of bone fracture, employing an inverse method based on a genetic algorithm. This experimental work considered hydrated and dehydrated cortical bovine bone tissue aiming to assess the influences of water on the cohesive laws and fracture toughness. The authors also pointed for the changes in material ductility and its effects on the fracture mechanisms (Figure 2.2). Silva et al. [36] applied a direct method to obtain the cohesive law for human cortical bone tissue under pure mode I loading using the DCB test. The direct method requires crack open displacement (w) monitoring using DIC and its correlation with the evolution of the strain energy release rate, GI (Figure 2.3a). The cohesive law is obtained by differentiation of this relation. A trapezoidal bilinear softening law was adjusted to the ensuing cohesive law (Figure 2.3b) and the resulting numerical load-displacement and

R-curves are compared with the experimental ones (Figure 2.3c, d). The observed

agreement validates all the procedure regarding the determination of cohesive laws under mode I loading of cortical bone tissue.

Figure 2.2 Crack propagation revealing extensive fibre-bridging [36].

0.3 mm

x, L z, T